Solution Manual for Applied Calculus 7th Edition by Waner Full file at https://./Solution-Manual-for-Applied-Calculus-7th-Edition-by-Wane SSoolluuttiioonnss SSeeccttiioonn 1 1 SSeeccttiioonn 1 1 Using the table:         a. 𝑓(0) = 2          b. 𝑓(2) = −0.5  Using the table:         a. 𝑓(−1) = 4          b. 𝑓(1) = −1  Using the table:         a. 𝑓(2) − 𝑓(−2) = −0.5 − = −2.5          b. 𝑓(−1)𝑓(−2) = (4)(2) = 8          c. −2𝑓(−1) = −2(4) = −8  Using the table:         a. 𝑓(1) − 𝑓(−1) = −1 − = −5          b. 𝑓(1)𝑓(−2) = (−1)(2) = −2          c. 3𝑓(−2) = 3(2) = 6  From the graph, we estimate:         a. 𝑓(1) = 20          b. 𝑓(2) = 30  In a similar way, we find:        c. 𝑓(3) = 30          d. 𝑓(5) = 20  e. 𝑓(3) − 𝑓(2) = 30 − 30 = 0          f. 𝑓(3 − 2) = 𝑓(1) = 20  From the graph, we estimate:         a. 𝑓(1) = 20          b. 𝑓(2) = 10  In a similar way, we find:         c. 𝑓(3) = 10          d. 𝑓(5) = 20  e. 𝑓(3) − 𝑓(2) = 10 − 10 = 0          f. 𝑓(3 − 2) = 𝑓(1) = 20  From the graph, we estimate:         a. 𝑓(−1) = 0          b. 𝑓(1) = −3  since the solid dot is on (1, −3) In a similar way, we estimate c. 𝑓(3) = 3  d. Since 𝑓(3) = 3  and 𝑓(1) = −3,   𝑓(3) − 𝑓(1) − (−3) = = 3.  3−1 3−1 From the graph, we estimate:         a. 𝑓(−3) = 3          b. 𝑓(−1) = −2  since the solid dot is on (−1, −2) Full file at https://./Solution-Manual-for-Applied-Calculus-7th-Edition-by-Waner Solution Manual for Applied Calculus 7th Edition by Waner Full file at https://./Solution-Manual-for-Applied-Calculus-7th-Edition-by-Wane SSoolluuttiioonnss SSeeccttiioonn 1 1 In a similar way, we estimate c. 𝑓(1) = 0  d. Since 𝑓(3) = 2  and 𝑓(1) = 0,   𝑓(𝑥) = 𝑥 − ,   with its natural domain 𝑥2 𝑓(3) − 𝑓(1) − = = 1.  3−1 3−1 The natural domain consists of all 𝑥 for which 𝑓(𝑥)  makes sense: all real numbers other than a. Since is in the natural domain, 𝑓(4)  is defined, and 𝑓(4) = − b. Since is not in the natural domain, 𝑓(0)  is not defined c. Since −1 is in the natural domain, 𝑓(−1) = −1 − 10 𝑓(𝑥) = − 𝑥 2,   with domain [2, +∞) 𝑥 a. Since is in [2, +∞), 𝑓(4)  is defined, and 𝑓(4) = (−1) = −1 − 1 63   =4− = 16 16 = −2.  31 − = − 16 = −   2 b. Since is not in [2, +∞), 𝑓(0)  is not defined         c.  Since is not in [2, +∞), 𝑓(1)  is not defined 11 𝑓(𝑥) = √𝑥 + 10,   with domain [−10, 0) a. Since is not in [−10, 0), 𝑓(0)  is not defined         b. Since is not in [−10, 0), 𝑓(9)  is not defined c. Since −10 is in [−10, 0), 𝑓(−10)  is defined, and 𝑓(−10) = √−10 + 10 = √0 = 0  12 𝑓(𝑥) = √9 − 𝑥 2,   with domain (−3, 3) a. Since is in (−3, 3), 𝑓(0)  is defined, and 𝑓(0) = √9 − = 3.  b. Since is not in (−3, 3), 𝑓(3)  is not defined         c. Since −3 is not in (−3, 3), 𝑓(−3)  is not defined 13 𝑓(𝑥) = 4𝑥 − 3  a. 𝑓(−1) = 4(−1) − = −4 − = −7          b. 𝑓(0) = 4(0) − = − = −3  c. 𝑓(1) = 4(1) − = − = 1          d. Substitute 𝑦 for 𝑥 to obtain 𝑓(𝑦) = 4𝑦 − 3  e. Substitute (𝑎 + 𝑏) for 𝑥 to obtain 𝑓(𝑎 + 𝑏) = 4(𝑎 + 𝑏) − 3.  14 𝑓(𝑥) = −3𝑥 + 4  a. 𝑓(−1) = −3(−1) + = + = 7          b. 𝑓(0) = −3(0) + = + = 4  Full file at https://./Solution-Manual-for-Applied-Calculus-7th-Edition-by-Waner Solution Manual for Applied Calculus 7th Edition by Waner Full file at https://./Solution-Manual-for-Applied-Calculus-7th-Edition-by-Wane SSoolluuttiioonnss SSeeccttiioonn 1 1 c. 𝑓(1) = −3(1) + = −3 + = 1          d. Substitute 𝑦 for 𝑥 to obtain 𝑓(𝑦) = −3𝑦 + 4  e. Substitute (𝑎 + 𝑏) for 𝑥 to obtain 𝑓(𝑎 + 𝑏) = −3(𝑎 + 𝑏) + 4.  15 𝑓(𝑥) = 𝑥 + 2𝑥 + 3  a. 𝑓(0) = (0) + 2(0) + = + + = 3          b. 𝑓(1) = + 2(1) + = + + = 6  c. 𝑓(−1) = (−1) + 2(−1) + = − + = 2          d. 𝑓(−3) = (−3) + 2(−3) + = − + = 6  e. Substitute 𝑎 for 𝑥 to obtain 𝑓(𝑎) = 𝑎 + 2𝑎 + 3.          f. Substitute (𝑥 + ℎ) for 𝑥 to obtain 𝑓(𝑥 + ℎ) = (𝑥 + ℎ) + 2(𝑥 + ℎ) + 3.  16 𝑔(𝑥) = 2𝑥 − 𝑥 + a. 𝑔(0) = 2(0) − + = − + =         b. 𝑔(−1) = 2(−1) − (−1) + = + + = c. Substitute 𝑟 for 𝑥 to obtain 𝑔(𝑟) = 2𝑟 − 𝑟 + d. Substitute (𝑥 + ℎ) for 𝑥 to obtain 𝑔(𝑥 + ℎ) = 2(𝑥 + ℎ) − (𝑥 + ℎ) + 17 𝑔(𝑠) = 𝑠 + a. 𝑔(1) = + c. 𝑔(4) = + 𝑠 1 = + =         b. 𝑔(−1) = (−1) + =1−1=0 −1 1 65 or 16.25         d. Substitute 𝑥 for 𝑠 to obtain 𝑔(𝑥) = 𝑥 + = 16 + = 4 𝑥 e. Substitute (𝑠 + ℎ) for 𝑠 to obtain 𝑔(𝑠 + ℎ) = (𝑠 + ℎ) + 𝑠+ℎ f. 𝑔(𝑠 + ℎ) − 𝑔(𝑠) =  Answer to part ( e)   −  Original function  = (𝑠 + ℎ) + ( 1 − 𝑠2 + 𝑠 + ℎ) ( 𝑠) 18 ℎ(𝑟) = 𝑟+4 c. ℎ(−5) = 1 = = −1         d. Substitute 𝑥 for 𝑟 to obtain ℎ(𝑥 2) = (−5) + (−1) 𝑥2 + a. ℎ(0) = 1 1 =         b. ℎ(−3) = = =1 0+4 (−3) + e. Substitute (𝑥 + 1) for 𝑟 to obtain ℎ(𝑥 + 1) = f. ℎ(𝑥 2) + = Answer to part (d) + = 1 = (𝑥 + 1) + 𝑥 + +1 𝑥2 + Full file at https://./Solution-Manual-for-Applied-Calculus-7th-Edition-by-Waner Solution Manual for Applied Calculus 7th Edition by Waner Full file at https://./Solution-Manual-for-Applied-Calculus-7th-Edition-by-Wane 19 𝑓(𝑥) = −𝑥 3  (domain (−∞, +∞)) Technology formula: -(x^3) 21 𝑓(𝑥) = 𝑥 4  (domain (−∞, +∞)) Technology formula: x^4   (𝑥 ≠ 0) 𝑥2 Technology formula: 1/(x^2) 23 𝑓(𝑥) = SSoolluuttiioonnss SSeeccttiioonn 1 1 20 𝑓(𝑥) = 𝑥 3  (domain [0, +∞)) Technology formula: x^3 22 𝑓(𝑥) = √𝑥  (domain (−∞, +∞)) Technology formula: x^(1/3) 24 𝑓(𝑥) = 𝑥 +   (𝑥 ≠ 0) 𝑥 Technology formula: x+1/x 25 a. 𝑓(𝑥) = 𝑥  (−1 ≤ 𝑥 ≤ 1)  Since the graph of 𝑓(𝑥) = 𝑥  is a diagonal 45° line through the origin inclining up from left to right, the correct graph is (A) b. 𝑓(𝑥) = −𝑥  (−1 ≤ 𝑥 ≤ 1)  Since the graph of 𝑓(𝑥) = −𝑥  is a diagonal 45° line through the origin inclining down from left to right, the correct graph is (D) c. 𝑓(𝑥) = √𝑥  (0 < 𝑥 < 4)  Since the graph of 𝑓(𝑥) = √𝑥  is the top half of a sideways parabola, the correct graph is (E) d. 𝑓(𝑥) = 𝑥 + − 2  (0 < 𝑥 < 4)  𝑥 Full file at https://./Solution-Manual-for-Applied-Calculus-7th-Edition-by-Waner Solution Manual for Applied Calculus 7th Edition by Waner Full file at https://./Solution-Manual-for-Applied-Calculus-7th-Edition-by-Wane SSoolluuttiioonnss SSeeccttiioonn 1 1 If we plot a few points like 𝑥 = 1/2, 1, 2, and 3, we find that the correct graph is (F) e. 𝑓(𝑥) = |𝑥|  (−1 ≤ 𝑥 ≤ 1)  Since the graph of 𝑓(𝑥) = |𝑥|  is a "V"-shape with its vertex at the origin, the correct graph is (C) f. 𝑓(𝑥) = 𝑥 − 1  (−1 ≤ 𝑥 ≤ 1)  Since the graph of 𝑓(𝑥) = 𝑥 − 1  is a straight line through (0, −1) and (1, 0), the correct graph is (B) 26 a. 𝑓(𝑥) = −𝑥 + 3  (0 < 𝑥 ≤ 3)  Since the graph of 𝑓(𝑥) = −𝑥 + 3  is a straight line inclining down from left to right, the correct graph must be (D) b. 𝑓(𝑥) = − |𝑥|  (−2 < 𝑥 ≤ 2)  Since 𝑓(𝑥) = − |𝑥|  is obtained from the graph of 𝑦 = |𝑥| by flipping it vertically (the minus sign in front of |𝑥|) and then moving it units vertically up (adding to all the values), the correct graph is (F) c. 𝑓(𝑥) = √𝑥 + 2  (−2 < 𝑥 ≤ 2)  The graph of𝑓(𝑥) = √𝑥 + 2  is similar to that of 𝑦 = √𝑥, which is half a parabola on its side, and the correct graph is (A) d. 𝑓(𝑥) = −𝑥 + 2  (−2 < 𝑥 ≤ 2)  The graph of 𝑓(𝑥) = −𝑥 + 2  is a parabola opening down, so the correct graph is (C) e. 𝑓(𝑥) = − 1  𝑥 The graph of 𝑓(𝑥) = − 1  (0 < 𝑥 ≤ 3)  is part of a hyperbola, and the correct graph is (E) 𝑥 f. 𝑓(𝑥) = 𝑥 − 1  (−2 < 𝑥 ≤ 2)  The graph of 𝑓(𝑥) = 𝑥 − 1  is a parabola opening up, so the correct graph is (B) 27 Technology formula: 0.1*x^2 - 4*x+5 Table of values: 𝑥 𝑓(𝑥)  1.1 −2.6 −6.1 −5 𝑔(𝑥) 39.9 −4 30.3 −3 21.5 −2 13.5 5.5 6.5 −9.4 −12.5 −15.4 −18.1 −20.6 −22.9 28 Technology formula: 0.4*x^2-6*x-0.1 Table of values: 𝑥 −1 6.3 −0.1 29 Technology formula: (x^2-1)/(x^2+1) Table of values: 𝑥 0.5 1.5 2.5 3.5 4.5 10 −25 −5.7 −10.5 −14.5 −17.7 −20.1 7.5 8.5 9.5 10.5 ℎ(𝑥) −0.6000 0.3846 0.7241 0.8491 0.9059 0.9360 0.9538 0.9651 0.9727 0.9781 0.9820 30 Technology formula: (2*x^2+1)/(2*x^2-1) Table of values: Full file at https://./Solution-Manual-for-Applied-Calculus-7th-Edition-by-Waner Solution Manual for Applied Calculus 7th Edition by Waner Full file at https://./Solution-Manual-for-Applied-Calculus-7th-Edition-by-Wane 𝑥 −1 SSoolluuttiioonnss SSeeccttiioonn 1 1 𝑟(𝑥) 3.0000 −1.0000 3.0000 1.2857 1.1176 1.0645 1.0408 1.0282 1.0206 1.0157 1.0124 𝑥       if  − ≤ 𝑥 <   { 2      if 0 ≤ 𝑥 ≤ Technology formula: x*(x=0) (For a graphing calculator, use ≥ instead of >=.) 31 𝑓(𝑥) = y -4 x a. 𝑓(−1) = −1.  We used the first formula, since −1 is in [−4, 0) b. 𝑓(0) = 2.  We used the second formula, since is in [0, 4] c.  𝑓(1) = 2.  We used the second formula, since is in [0, 4] −1      if  − ≤ 𝑥 ≤   { 𝑥       if 0 < 𝑥 ≤ Technology formula: (-1)*(x0) (For a graphing calculator, use ≤ instead of